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Some Special Types of Orbits Around Jupiter aerospace Article Some Special Types of Orbits around Jupiter Yongjie Liu 1, Yu Jiang 1,*, Hengnian Li 1,2,* and Hui Zhang 1 1 State Key Laboratory of Astronautic Dynamics, Xi’an Satellite Control Center, Xi’an 710043, China; [email protected] (Y.L.); [email protected] (H.Z.) 2 School of Electronic and Information Engineering, Xi’an Jiaotong University, Xi’an 710049, China * Correspondence: [email protected] (Y.J.); [email protected] (H.L.) Abstract: This paper intends to show some special types of orbits around Jupiter based on the mean element theory, including stationary orbits, sun-synchronous orbits, orbits at the critical inclination, and repeating ground track orbits. A gravity model concerning only the perturbations of J2 and J4 terms is used here. Compared with special orbits around the Earth, the orbit dynamics differ greatly: (1) There do not exist longitude drifts on stationary orbits due to non-spherical gravity since only J2 and J4 terms are taken into account in the gravity model. All points on stationary orbits are degenerate equilibrium points. Moreover, the satellite will oscillate in the radial and North-South directions after a sufficiently small perturbation of stationary orbits. (2) The inclinations of sun-synchronous orbits are always bigger than 90 degrees, but smaller than those for satellites around the Earth. (3) The critical inclinations are no-longer independent of the semi-major axis and eccentricity of the orbits. The results show that if the eccentricity is small, the critical inclinations will decrease as the altitudes of orbits increase; if the eccentricity is larger, the critical inclinations will increase as the altitudes of orbits increase. (4) The inclinations of repeating ground track orbits are monotonically increasing rapidly with respect to the altitudes of orbits. Keywords: mean element theory; orbit dynamics; longitude drift; critical inclinations Citation: Liu, Y.; Jiang, Y.; Li, H.; Zhang, H. Some Special Types of Orbits around Jupiter. Aerospace 2021, 8, 183. https://doi.org/10.3390/ 1. Introduction aerospace8070183 Jupiter is the most massive planet in the solar system. It is also a gas giant planet. Its mass is 2.5 times the mass of other planets in the solar system. The large size of Jupiter also Academic Editor: Fanghua Jiang makes it relatively easy to be observed. As a result, it was discovered very early. Jupiter has been one of the major targets for planetary exploration. However, Jupiter’s powerful Received: 19 April 2021 magnetosphere and radiation belts are threats to all human spacecraft trying to visit Jupiter. Accepted: 4 July 2021 Since the 1970s, several space missions have been launched by NASA, such as Pioneer X, Published: 8 July 2021 Voyager 1, Galileo, and Juno. However, there is still a long way to go to explore Jupiter. There exist many interesting phenomena that have attracted people to explore for many Publisher’s Note: MDPI stays neutral years, for example, the Great Red Spot, Jupiter’s rings, and atmospheric jet streams. As with regard to jurisdictional claims in a gaseous planet, Jupiter cannot be explored by landing like a lithospheric planet. We published maps and institutional affil- can only use probes to orbit and enter Jupiter’s atmosphere. Therefore, it is necessary to iations. investigate the orbit dynamics around Jupiter. Weibel et al. [1] researched stable orbits between Jupiter and the Sun. Jacobson [2] investigated the gravity field of Jupiter and the orbits of its Galilean satellites. Colwell et al. [3] studied the exogenic dust ring. Recently, Liu et al. [4,5] discussed the dust in the Jupiter system outside the rings and distribution of Copyright: © 2021 by the authors. Jovian dust ejected from the Galilean satellites. Research about this will surely contribute Licensee MDPI, Basel, Switzerland. to the orbit design of space missions. This article is an open access article Investigating the dynamical environments around planets has been the focus for distributed under the terms and space missions in the past few decades. For this purpose, much research concerning conditions of the Creative Commons various special artificial satellite orbits around planets have been conducted. Usually, these Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ special types of orbits include stationary orbits, frozen orbits, sun-synchronous orbits, 4.0/). repeating ground-track orbits, and orbits at the critical inclination. The original idea about Aerospace 2021, 8, 183. https://doi.org/10.3390/aerospace8070183 https://www.mdpi.com/journal/aerospace Aerospace 2021, 8, 183 2 of 15 geostationary orbits was first put up by Clarke [6]. He pointed out that satellites with an altitude of 36,000 km above the equator of the Earth would have the same rotation rates as the Earth and stay stationary relative to an observer on the equator. Therefore, geostationary satellites are often used for the sake of communications and navigations. Four equilibrium solutions for geostationary orbits were shown to exist by Musen & Bailie [7]. Moreover, two of them were stable while the other two were unstable. Lara & Elipe [8] calculated periodic orbits around equilibrium points in the Earth second degree and order gravity field. For Mars, the stationary orbits, also known as areostationary orbits, and the equilibrium points were studied by Liu et al. [9]. The periodic orbits around the equilibrium points were also calculated by Liu et al. [10]. For orbits at the critical inclination, the eccentricity and argument of perigee are invariant on average. The concept of the Earth critical inclination was first introduced by Orlov [11]. Brouwer [12] used canonical transformations to eliminate short-period terms. Coffey et al. gave a geometrical interpretation of the critical inclination for satellites by investigating the averaged Hamilton system [8]. Representatives of orbits at the critical inclination are the Russian Molniya satellites. The combined effects of the critical inclination and the 2:1 mean motion resonance of a Molniya orbit have been intensively studied since then, for example, in [13–17]. Similarly, frozen orbits are characterized by the invariance of average eccentricity and argument of perigee. Frozen orbits are not limited to specific inclinations. They may exist at any inclination. Usually, the argument of perigee is equal to 90 or 270 deg, depending on the sign of the ratio of the harmonic coefficients J3 and J2. Frozen orbits were first proposed by Cutting et al. [18] for orbit analysis of the Earth satellite SEASAT-A. Coffey et al. [19] showed that there exist three families of frozen orbits in the averaged zonal problem up to J9 in the gravity field of an Earth-like planet. The frozen orbits around the moon in the full gravity model were considered by Folta & Quinn [20], and Nie & Gurfil [21]. Some researchers also view orbits at the critical inclination as frozen orbits, for example [19,22]. Sun-synchronous orbits are defined with a precession rate of the orbital plane equal to the revolution angular velocity around the sun. Generally, remote sensing satellites are placed into these orbits. Macdonald et al. [23] used an undefined, non-orientation- constrained, low-thrust propulsion system to consider an extension of the sun-synchronous orbits. For repeating ground track orbits, the trajectory ground track repeats after a whole number of revolutions within some days. Orbits of this type are widely used to achieve better coverage properties. Lara showed that orbits repeating their ground track on the surface of the Earth were members of periodic-orbit families of the tesseral problem of the Earth artificial satellite [24]. Lei [25] considered the leading terms of the Earth’s oblateness and the luni-solar gravitational perturbations to describe the secular dynamics of navigation satellites moving in the medium Earth orbit and geosynchronous orbit regions. Liu et al. [9] calculated these five types of special orbits around Mars with analytical formulations and numerical simulations. In fact, the gravity field of Mars shares many similarities with that of the Earth. The J2 terms of them are dominant among the harmonic coefficients. However, the J2 term is not as dominant as Earth’s J2. The other first few harmonic coefficients are also strong for Mars: about 1–2 orders of magnitude smaller than J2; for the Earth, the other first few harmonic coefficients are about 3–4 orders of magnitude smaller than J2. The situation is rather different for Jupiter compared with the Earth and Mars. Due to the difficulty of determining the gravity field of Jupiter, there exist few studies about special types of orbits around Jupiter, as far as we know. However, the situation has greatly improved since Juno’s gravitational measurements were conducted. Iess et al. [26] provided measurements of Jupiter’s gravity harmonics (both even and odd) through precise Doppler tracking of the Juno spacecraft. Moreover, they pointed out a North-South asymmetry, which is a signature of atmospheric and interior flows. Here, we mainly use the results Aerospace 2021, 8, 183 3 of 15 in [26] to build a simplified gravity model of Jupiter. Some harmonic coefficients of the gravity model of the Earth, Mars, and Jupiter can be seen in Table1. Table 1. Some harmonic coefficients and their ratios of the Earth, Mars, and Jupiter. −3 −6 −5 −6 −3 −2 Planet J2(×10 ) J3(×10 ) J4(×10 ) J22(×10 ) J3/J2(×10 ) J4/J2(×10 ) Earth ([27]) 1.08263 −2.53266 −1.61962 1.81534 −2.3394 −1.4960 Mars ([28]) 1.95545 31.4498 −1.53774 63.0692 16.083 −0.7864 Jupiter ([26]) 14.6965 0.042 −58.661 0 0.0021378 −3.9923 In this paper, we investigate some special orbits around Jupiter, considering mainly the effect of the non-spherical perturbation of the gravity field. From Table1, one can see that the J2 term is still dominating, J2 is about 25 times the 5 value of J4, but is 10 times bigger than J3.
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